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Poker - Probability of Poker Hands Drew Armstrong...

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Unformatted text preview: Probability of Poker Hands Drew Armstrong [email protected] November 1, 2006 In a standard deck of cards, there are 4 possible suits (clubs, diamonds, hearts, spades), and 13 possible values (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Ace). Let A,J,Q,K represent Ace, Jack, Queen and King, respectively. Every card has a suit and value, and every combination is possible. Hence a standard deck contains 13 · 4 = 52 cards. A “poker hand” consists of 5 unordered cards from a standard deck of 52. There are ( 52 5 ) = 2 , 598 , 9604 possible poker hands. Below, we calculate the probability of each of the standard kinds of poker hands. Royal Flush. This hand consists of values 10 ,J,Q,K,A , all of the same suit. Since the values are fixed, we only need to choose the suit, and there are ( 4 1 ) = 4 ways to do this. Straight Flush. A straight flush consists of five cards with values in a row, all of the same suit. Ace may be considered as high or low, but not both. (For example, A, 2 , 3 , 4 , 5 is a straight, but Q,K,A, 2 , 3 is not a straight.) The lowest value in the straight may be A, 2 , 3 , 4 , 5 , 6 , 7 , 8 or 9. (Note that a straight flush beginning with 10 is a royal flush, and we don’t want to count those.) So there are 9 choices for the card values, and then ( 4 1 ) = 4 choices for the suit, giving a total of 9 · 4 = 36....
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